 Solution

The capital asset pricing model givesThus it has the same affine relation between expected returns and a single systematic exposure. When the factor is the market portfolio's excess return, the loadings are the beta of an asset, ${\lambda }_0=r_f$, and ${\lambda }_1=\mathbb{E}{r}_M-r_f$. More generally, for a nondegenerate exact one-factor model and a market portfolio with loading $b_M\ne 0$, ${\beta }_i=b_i/b_M$, and the pricing relation can be rewritten using that exposure.

The interpretations and assumptions differ. The CAPM is an equilibrium relation obtained with a mean-variance optimization framework, common beliefs, and suitable borrowing/lending and market-clearing assumptions. Arbitrage pricing theory uses absence of arbitrage, factor structure, and in realistic models diversification; it does not identify a generic factor with the market portfolio. Moreover the CAPM permits asset-specific risk, whereas the exact model in part (a) has no residual risk. The two formulas coincide for a suitable market factor, but no-arbitrage factor pricing by itself does not establish the CAPM's equilibrium assumptions.